Can you fill this chessboard?
I am a big fan of riddles, especially those which are extremely simple to explain while having an elegant solution. So, here it is one of my favorites!
Imagine I give you a chessboard with its classical 8×8 squares, but without the top left and bottom right corners (see image below).
Question: can you fill the entire board with pieces of dominoes?
Note: each domino occupies two adjacent squares, so diagonals are not allowed.
More than 15 years ago, when I was a naïve young teenager, I discovered this riddle in the book Fermat’s last theorem by Simon Singh (I highly recommend it!). This problem was an example of a mathematical proof without using brute force. The simplicity of the solution just blew my mind. Since then, I became an amateur riddle collector. I hope you also enjoy solving this little problem!
Our first instinct is to start try out possibilities because, if we find only one case, then we show that it is possible to fill the entire board. However, you will quickly find out that it isn’t trivial. Something is wrong and you cannot really tell why. So, let me tell you that it isn’t possible to fill the board. The trick is to look at the squares that a domino occupies: one black and one white. Since we have removed two white squares, a domino would need to occupy two black ones, which is impossible.
Oh gosh. I love this solution. I hope you solved it by yourself! Just for curiosity, while writing this post, I found out that this riddle is known as Mutilated chessboard problem (poor board!).